Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $\overline {T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline {T\pi (g)}$ is homogeneous (i.e. $\overline {T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

Topological generation results for free unitary and orthogonal groups

We show that for every [Formula: see text] the free unitary group [Formula: see text] is topologically generated by its classical counterpart [Formula: see text] and the lower-rank [Formula: see text]. This allows for a uniform inductive proof that a number of finiteness properties, known to hold for all [Formula: see text], also hold at [Formula: see text]. Specifically, all discrete quantum duals [Formula: see text] and [Formula: see text] are residually finite, and hence also have the Kirchberg factorization property and are hyperlinear. As another consequence, [Formula: see text] are topologically generated by [Formula: see text] and their maximal tori [Formula: see text] (dual to the free groups on [Formula: see text] generators) and similarly, [Formula: see text] are topologically generated by [Formula: see text] and their tori [Formula: see text].

On Splitting of the Normalizer of a Maximal Torus in E6(q)

Let G be a finite group of Lie type E6 over 𝔽q (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∊ W such that N(G, T )/T ≃ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.

MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.